Section 6 Charts for Multiple Locations

Lesson S06-02

Text: Section 6 pages 8 – 13

Duration: 11 minutes

Introduction to X̄, Rb, and S Charts

Whenever the measurements in a sample come from the same distribution, and the sample to sample variation is much bigger than the within sample variation, Rb and S charts should be used.

An Example

Calculations for the X̄, Rb, and S Charts

Download the worksheet for your own use:

X̄, Rb, and S Charts Worksheet

Making Sense of the Calculations

X̄, Rb, and S Charts Minitab Example

An Example of Rational Sample Violation

Section 6 Charts for Multiple Locations

Lesson S06-01

Text: Section 6 pages 1 – 7

Duration: 17 minutes

Introduction to Multiple Locations Charts

There are many instances where subgroups of size two or more are most convenient, but there is a possible violation of rational sampling. For example, a workstation machines several parts per cycle, so that the parts in the subgroup were machined in the same cycle, but at different locations on the workstation.

Another instance might be that there are multiple production lines performing the same task. Another example is destructive testing, where multiple measurements are often taken on a single part (to avoid destruction of multiple parts), but the part might not be homogeneous.

In any case, traditional SPC does not handle the situations described above unless the locations are essentially the same.

A Possible Solution

Still a Problem

An Example of the Problem

A Plot to Detect the Problem

Section 5 Short Run SPC

Lesson S05-04

Text: Section 5 page 17 – Section 9 page 18

Duration: 29 minutes

Introduction to Exercise 11

Data were collected from 2 different parts produced with the same process. The sample size is 2. Part A: Nominal Value = 5.20 Part B: Nominal Value = 8.00 The data can be found in Exercise11.MTW and Exercise11.XLS.

a. Construct a DNOM Chart. Is the DNOM Chart appropriate for this data?

Solution / Discussion of Exercise 11 a

b. Conduct a Test of Equal Variances for the data from the 2 parts

Solution / Discussion of Exercise 11 b

c. Construct Standardized DNOM and Standardized S Charts.

Solution / Discussion of Exercise 11 c

Section 5 Short Run SPC

Lesson S05-03

Text: Section 5 pages 13 – 16

Duration: 11 minutes

Using Minitab to Construct Standardized DNOM and S Charts

MINITAB does not provide direct support for the charts for short production runs (DNOM, Standardized DNOM, Standardized S). However, once the relevant statistics are computed and control limits determined, the “Individuals Chart” functionality may be used to construct the appropriate charts.

Section 5 Short Run SPC

Lesson S05-02

Text: Section 5 pages 4 – 12

Duration: 18 minutes

Handling Standard Deviation Differences

As discussed above, when the part types do not have similar standard deviations, the DNOM method and charts are invalid. To account for differences in variation between the part types, a Standardized DNOM (to monitor location) and Standardized S chart (to monitor variation) may be created.

Next, we briefly illustrate a procedure to test for statistical differences in variation among groups. The technical details are left for a course in hypothesis testing.

Standardized DNOM Chart Data

Standardized DNOM Chart Setup

Section 5 Short Run SPC

Lesson S05-01

Text: Section 5 pages 1 – 4

Duration: 15 minutes

Introduction to Short Run Charts

Traditional SPC methods were developed to support high volume production and long production runs. However, with the trend toward product specialization, product diversity, and flexible manufacturing, short production runs have become more common. Applying SPC in the traditional manner presents challenges in short production runs, because by the time enough data is collected to establish valid control charts, the production run may be over! An approach to deploying control charts with short production runs is to utilize charts of common characteristics across different products. The chart pertains to the characteristic of interest (e.g. diameter) rather than for the diameter of a specific design.

Introduction to DNOM Charts

A single chart can monitor the characteristic even though the nominal values (and specifications) are different. This is accomplished by standardizing the data before plotting it. A common chart that performs this is called the Deviations from Nominal (DNOM) chart. Essentially, the value that is plotted is the difference between the part measurement and the nominal value (Note: The nominal value refers to the value that is typically specified on an engineering drawing as the desired value and is often halfway between the lower and upper specification limits). Once the differences are computed, the control limits may be established in the normal way.

A DNOM Charting Example

DNOM Assumptions & Considerations

Section 4 Process Capability

Lesson S04-11

Text: Section 4 pages 48

Duration: 2 minutes

Process Capability Section Summary

Section 4 Process Capability

Lesson S04-10

Text: Section 4 pages 46 & Section 9 pages 16 – 17

Duration: 15 minutes

Exercise 10

Assess the process capability (using MINITAB) of the O-ring parting line extension data in file Exercise10.MTW. The suggested steps are below. The upper specification limit is 0.08 mm

a. Test the data for normality. What are your findings?

b. Perform a natural log transformation on the data and test the transformed data for normality. What are your findings?

c. Estimate the ppm and Ppk using the transformed data. Remember to also transform the specification limit.

d. Attempt to find a reasonable distribution to describe the original data. Try the Weibull, Lognormal, Gamma, and Loglogistic distributions. What are your conclusions?

e. Use an appropriate distribution to estimate the ppm and Ppk (use MINITAB’s Non-normal process capability option).

Solution / Discussion of Exercise 10

Section 4 Process Capability

Lesson S04-09

Text: Section 4 pages 46

Duration: 12 minutes

Distribution Fitting Minitab Example

In MINITAB, distribution fitting may be performed using the Individual Distribution Identification option under Quality Tools. Just as we tested for normality using a Normal Probability Plot, we can use probability plots for other distributions. Here, the more closely the data points appear linear on specific probability paper, the better the fit.

What if We Just Assume Normality?

The Johnson Transformation

Section 4 Process Capability

Lesson S04-08

Text: Section 4 pages 45

Duration: 19 minutes

A Non-Normal Minitab Example

After determining an appropriate distribution to model the nonnormal data, we just learned how to compute percentiles (in order to correctly estimate capability indices). Statistical software such as MINITAB simplifies the steps needed to estimate process capability of nonnormal data.

A Minitab Transformation Example

Another Transformation Option

Section 4 Process Capability

Lesson S04-07

Text: Section 4 pages 33 – 44

Duration: 33 minutes

Testing for Normality

So far, we have assumed that the individual data values are normally distributed. In practice, we need to test this assumption before using methods for normal data. Using methods for normal data on non-normal data will produce misleading (often too optimistic) capability estimates.

Here, we briefly describe how to perform a normality test on the data. We hypothesize that our data follows a normal distribution, and only reject this hypothesis if we have strong evidence to the contrary.

Normal Probability Plotting

Normal probability plotting may be used to objectively assess whether data comes from a normal distribution, even with small sample sizes. On a normal probability plot, data that follows a normal distribution will appear linear (follow a fairly straight line). For example, a random sample of 30 data points from a normal distribution results in the normal probability plot below.

Handling Non-normal Data

This introductory course primarily focuses on estimating process capability for normally distributed data. Methods for handling nonnormal data are briefly discussed here.

Transformations

Data transformations may be performed which will cause the transformed data to be normally distributed. Taking the log (or natural log) of the data is a common choice. This transformation tends to make skewed data appear more bell-shaped because the log function takes large numbers and brings them back “into the pack.” The smaller numbers are also transformed but they are not affected as much as the larger numbers.

Distribution Fitting

Another method for handling nonnormal data is to try to find a distribution that describes or “fits” the data. Practically speaking, this approach requires statistical software that allows multiple distributions to be fit to the data.

If a reasonable fit is found for a known distribution, then we can utilize the software to compute the required percentiles for our procedure. Some distributions are very flexible and can assume a wide variety of shapes depending on the specified parameters. For example, the Weibull distribution is a commonly used distribution due to its flexibility.

Section 4 Process Capability

Lesson S04-06

Text: Section 4 pages 32 & Section 9 page 14 – 15

Duration: 16 minutes

Introduction to Exercises 8 & 9

A Quick Capability Analysis Minitab Tutorial

Exercise 8

For the Bended Rubber data in the previous problem, compute the Pp and Ppk. Note that since the sample standard deviation (s) was provided, the appropriate indices are Pp and Ppk. (Cpk is based on the range estimator for the standard deviation).

Summarized statistics from the data and specification limits follow:

• Average (X-Bar) = 69.67
• Std. Deviation (s) = 1.63
• USL = 75.0
• LSL = 65.0

Note: The data has been shown to follow a normal distribution.

Solution / Discussion of Exercise 8

Exercise 9

For the Bended Rubber data in the previous problem, use MINITAB to estimate the Ppk. Also, verify that the process is stable by constructing Individuals and Moving Range charts.

The data may be found in file, Exercise09.MTW.

Solution / Discussion of Exercise 9

Section 4 Process Capability

Lesson S04-05

Text: Section 4 pages 22 – 31

Duration: 38 minutes

Introduction to Cpk

In order to assess two distinct features of the process, we need two estimates–one for location and one for variation. Unfortunately, this simple notion has been ignored, and many have the false impression that a Cpk can describe both aspects of the process distribution. Rather, the Cpk is just a Cp for the worst side of the distribution of individuals. We review the details next.

Calculation of Cpk

Cp & Cpk Discussion

Cp & Cpk Assumptions

Cp & Cpk & PPM

A Minitab Example

Section 4 Process Capability

Lesson S04-04

Text: Section 4 pages 17 – 21

Duration: 12 minutes

Introduction to Capability Indices

The Calculation of Cp

Answer and Another Quick Quiz

Answer and Summary of Cp

Section 4 Process Capability

Lesson S04-03

Text: Section 4 pages 16 & Section 9 page 13

Duration: 9 minutes

Introduction to Exercise 7

Batches of Blended Rubber to be used for O-Rings are tested with regard to hardness. Samples were taken and tested from 50 consecutive batches. The process has been shown to be stable and the individual values are normally distributed.

Summarized statistics from the data and specification limits follow:

• Average (X-Bar) = 69.67
• Std. Deviation (s) = 1.63
• USL = 75.0
• LSL = 65.0

Estimate the ppm (expected number of defective O-Rings per million produced)

Solution / Discussion of Exercise 7